Geometry: Proof by Contradiction: The Advantage of Being Indirect
Proof by Contradiction: The Advantage of Being Indirect
You have already observed that a conditional statement is logically equivalent to its contrapositive. It is reasonable to expect that if one logical version of a statement is a tautology, so is another. This is an important point because there are times when writing a direct proof to a theorem is awkward and difficult. When that happens, you might have an easier time if you turn to the negative side.
For example, suppose that if the power fails, you will be late for work. What conclusion can you draw if you were not late for work? For one thing, you know that the power must not have failed. Why? Well, the statement “the power fails” is either true or false (there is no other option). If the statement “the power fails” was true, then by your premise you would have been late for work. Because you were not late for work, it must be false that the power failed. This type of argument is called an indirect argument. You managed to make your argument through the back door, so to speak. You assumed an opposite conclusion, and contradicted your premises.
In an indirect proof of a theorem P → Q, you assume P → Q and ~ Q, and try to show ~ P. You can write this as ((P → Q) ∧ ~ Q) →~ P, and construct a truth table for this compound statement.
| Truth table for ((P → Q) ∧ ~ Q) → ~ P | ||||||
|---|---|---|---|---|---|---|
| P | Q | P → Q | ~ Q | (P → Q)∧ ~ Q | ~ P | ((P → Q) ∧ ~ Q) → ~ P |
| T | T | T | F | F | F | T |
| T | F | F | T | F | F | T |
| F | T | T | F | F | T | T |
| F | F | T | T | T | T | T |
Eureka!
The key to a proof by contradiction is that you assume the negation of the conclusion and contradict any of your definitions, postulates, theorems, or assumptions. You need to contradict something you either believe to be true or have defined to be a certain way.
Because ((P → Q) ∧ ~ Q) → ~ P is a tautology, you have yet another valid argument: Assume P → Q and ~ Q and show ~ P.
Indirect proofs are particularly useful when proving statements about parallel lines. This is because when two lines are parallel, they do not intersect. This is a rather negative definition (the criteria for our definition is met by the two lines failing to do something in particular), and by negating the negative you turn everything positive. This can make things easier for you.
Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.
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